Research in Scientific Computing in Undergraduate Education

Convergence of a sequence

Given a real number x, the sequence e(n,x) is defined recursively by:

e(1,x)=x,

e(2,x)=1,

e(n,x)=e(n-1,x)+e(n-2,x)/(n-2)\textrm{ for }n \geq 3.

(i) Does \frac{n}{e(n,x)} approach a limit as n increases without bound?

(ii) How does this limit, when it exists, depend on x?

(iii) When there is a limit, how rapidly does \frac{n}{e(n,x)} converge to the limit? Does this depend on x?

This is an investigation in analysis, which is quite hard without recourse to computational methods. It is a nice example of how computational techniques can provide information about a strange sequence.

More generally, what is the behavior of sequences e(n,x) defined recursively by:

e(1,x)=x,

e(2,x)=1,

e(n,x)=e(n-1,x)/(an-p)+e(n-2,x)/(bn-q)\textrm{ for }n \geq 3?

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